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The paper utilizes data from the SuperKamiokande solar neutrino detection experiment and analyses them by diffusion entropy analysis and standard deviation analysis. The result indicates that solar neutrinos are subject to Levy flights (anomalous diffusion, superdiffusion). Subsequently, the paper derives the probability density function, represented as Fox H-function, and the governing fractional diffusion equation (fractional Fokker-Planck Equation) for solar neutrino Levy flights., Comment: 13 pages. arXiv admin note: substantial text overlap with arXiv:1209.1520

The explicit forms of the exact density and the exact arbitrary moments of the distance between two random points when the points are independently and uniformly distributed inside a cube are given. Special cases considered include the density and arbitrary moments of the distance between two independent random points where one is inside the cube and the other is on a face, points on adjacent faces, points on opposite faces and random points inside a square. Exact arbitrary moments are obtained in terms of Gauss'hypergeometric functions and Appell's functions or Lauricella functions.

A multivariate beta model is introduced by combining univariate real beta variables and uniform random variables. Various properties of this model are studied, including product moments, conditional moments, exact density, Chebyshev's type inequalities and estimation of parameters.

One of the most popular test statistics is Pearson's c2 goodness-of-fit statistic which is known to have an approximate chi-square distribution for large sample size. All the proofs so far seem to make use of the chisquaredness of quadratic forms in normal variables, manipulations of quadratic forms and singular covariance matrix. An elegant simple derivation of this result is given here without making use of chisquaredness of quadratic forms or other such results. Two numerical examples are also given to caution about the use of this statistic.